Related papers: On the $H^1(ds)$-gradient flow for the length func…

We address the problem of steady laminar flow of a shear-thinning fluid in rectangular ducts, which is encountered in many systems, in particular, in microfluidic and biomedical devices. However, an exact solution for the flow of…

We propose a variational form of the BDF2 method as an alternative to the commonly used minimizing movement scheme for the time-discrete approximation of gradient flows in abstract metric spaces. Assuming uniform semi-convexity --- but no…

We explicitly construct parameter transformations between gradient flows in metric spaces, called curves of maximal slope, having different exponents when the associated function satisfies a suitable convexity condition. These…

We develop a geometric flow framework to investigate two classical shape functionals: the torsional rigidity and the first Dirichlet eigenvalue of the Laplacian. First, by constructing novel deformation paths governed by height-stretching…

This paper introduces new methods to study the long time behaviour of the generalised gradient flow associated with a solution of the critical equation for mechanical Hamiltonian system posed on the flat torus $\mathbb{T}^d$. For this…

The gradient flow is the evolution of fields and physical quantities along a dimensionful parameter~$t$, the flow time. We give a simple argument that relates this gradient flow and the Wilsonian renormalization group (RG) flow. We then…

In this paper we consider the evolution of regular closed elastic curves $\gamma$ immersed in $\R^n$. Equipping the ambient Euclidean space with a vector field $\ca:\R^n\rightarrow\R^n$ and a function $f:\R^n\rightarrow\R$, we assume the…

We study gradient flows of general functionals with linear growth with very weak assumptions. Classical results concerning characterisation of solutions require differentiability of the Lagrangian, as for the time-dependent minimal surface…

It is well-known that the mean curvature flow is a formal gradient flow of the perimeter functional. However, by the work of Michor and Mumford [7,8], the formal Riemannian structure that is compatible with the gradient flow structure…

Flows on the moduli space of the algebraic Riemann surfaces, preserving the periods of the corresponding solutions of the soliton equations are studied. We show that these flows are gradient with respect to some indefinite symmetric flat…

In this paper, we define a family of functionals generalizing the Yang-Mills-Higgs functional on a closed Riemannian manifold. Then we prove the short time existence of the corresponding gradient flow by a gauge fixing technique. The lack…

We study the geodesic flow corresponding to the left-invariant sub-Riemannian metric and the right-invariant distribution on the second Heisenberg group. The corresponding Hamiltonian system is completely integrable and in this paper we…

Recently Andrews and Bryan [3] discovered a comparison function which allows them to shorten the classical proof of the well-known fact that the curve shortening flow shrinks embedded closed curves in the plane to a round point. Using this…

We investigate a family of gradient flows of positive and probability measures, focusing on the Hellinger-Kantorovich (HK) geometry, which unifies transport mechanism of Otto-Wasserstein, and the birth-death mechanism of Hellinger (or…

In this work, we investigate links between the formulation of the flow of marginals of reversible diffusion processes as gradient flows in the space of probability measures and path wise large deviation principles for sequences of such…

We introduce a flow of $G_2$-structures defining the same underlying Riemannian metric, whose stationary points are those structures with divergence-free torsion. We show short-time existence and uniqueness of the solution.

We consider a general formulation of gradient flow evolution for problems whose natural framework is the one of metric spaces. The applications we deal with are concerned with the evolution of {\it capacitary measures} with respect to the…

On a compact Riemannian manifold, we study the various dynamical properties of the Schr\"odinger flow $(e^{it\Delta/2})$, through the notion of semiclassical measures and the quantum-classical correspondence between the Schr\"odinger…

Let $M$ be a closed Riemannian manifold with a parallel 1-form $\Omega$. We prove two theorems about the curve shortening flow in $M$. One is that the {\csf} $\ct$ in $M$ exists for all $t$ in $[0, \infty)$, if it satisfies $\Omega(T)\geq…

It is well-known that the class of piecewise smooth curves together with a smooth Riemannian metric induces a metric space structure on a manifold. However, little is known about the minimal regularity needed to analyze curves and…